The Magnitude of a Vector

This quantity has units of 1/length and is the magnitude of a vector known as the scattering vector. For our purpose here, it is sufficient to note that s is the yardstick used for measuring the distance between the scattering centers when the intensity of the scattered radiation of wavelength X is recorded at an angle 0 from the direction of the incident radiation. 

This was done in the previous chapter when determining the magnitude of a vector. 

A dimension in reciprocal space is a reciprocal of the dimension in real space (with a factor of unity). The magnitude of a vector d hkl in a reciprocal lattice equals the reciprocal of plane spacing (<4h) in real space. 

In the Nyquist format, the is plotted on they axis and Zreai is plotted on the x axis (Fig. 6.1a-d). The impedance magnitude, Z, for a circuit like the one shown in Fig. 6.1c is equal to the magnitude of a vector from the origin (0,0) to the point of interest (x,y) and the phase angle, (p, is given by the angle between the vector and the Zreai... 

The magnitude of a vector A is denoted by A or by A. The scalar product of a vector with itself gives the square of the magnitude of the vector ... 

If the force and the displacement are not in the same direction, they must be treated as vectors. A vector is a quantity that has both magnitude and direction. Vectors are discussed briefly in Appendix B. We denote vectors by boldface letters and denote the magnitude of a vector by the symbol for the vector between vertical bars or by the letter in plain type. The amount of work dw can be written as the scalar product of the two vectors F and dr where F is the force exerted on the object and dr is its displacement ... 

We use either of the two notations in Eq. (9.2-11) to denote the magnitude of a vector the boldface letter within vertical bars or the letter in plain type. The magnitude of a vector is always non-negative (positive or zero). 

For three orthogonal axes, v coordinates, s, q, f, are defined such that s is the magnitude of a vector in v, space, q is the elevation angle relative to the plane (q = 0 corresponds to... 

The magnitudes of a vector a and a second-order tensor x are defined by ... 

A vector whose magnitude is equal to the magnitude of A but is in the opposite direction is denoted by -A. 

However, the scalar magnitudes of g and/ from both antennas (G and F) are the same, because the scalar magnitude of a vector is the square root of the vector squared. Thus the following quantity is radiated into the vacuum ... 

The polar plot is an alternative to the Bode diagram for representing frequency response data and is the locus of all points occupied by the tip of a vector in the complex plane whose magnitude and direction are determined by the amplitude ratio and phase shift, respectively, as the frequency of the forcing function applied to the system is varied from zero to infinity. 

What current density vector y (units Am ) is produced by the application of an electric field of 10 V/m along the [1 1 1] direction of the crystal What is the angle between y and El What is the magnitude of a-along the [111] ... 

The concept of a vector, however, was originally introduced in physics applications to describe quantities having both magnitude and direction, such as force and velocity. Later, the concept was blended with many of the other notions of linear algebra when mathematicians realized that vectors and one column (or one row) matrices are mathematically identical. 

The dot product Jri x2 is the number obtained by multiplying the length (or magnitude) of one vector times that of the other, times the cosine of the angle between them when the tails are placed at a common point. The length of a vector multiplied by the cosine is the projection of that vector on the other. The dot product may also be written in component form as X] -x2 = (x]x2 + y y2 + zizi). 

X, Y and Z are the magnitudes of reciprocal vector R, corresponding to the magnitude (x, y and z) of vector r in the real space. Since the electron waves pass the objective lens, the objective lens can affect the wave characteristics. The effect of the objective lens on the wave function can be represented by a transfer function, 7TR). Thus, the actual wave function on the back-focal plane becomes the following function. 

Thus, the solubility parameter may be thought of as a vector in a three-dimensional d-p-h space. The above equation provides the magnitude of this vector. Each solvent and each polymer can be characterized by the three solubility parameter increments 5, 5, 5j. 

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